Problem 25 - Precalculus

Question

Assume that the coordinates of the points \(P\) \(Q, R, S,\) and \(O\) are as follows: \(P(-1,3) \quad Q(4,6) \quad R(4,3) \quad S(5,9) \quad O(0,0)\) Draw the indicated vector (using graph paper) and compute its magnitude. Compute the sums using the definition. Use the parallelogram law to compute the sums. $$\overrightarrow{S O}+\overrightarrow{S Q}$$

Step-by-Step Solution

Answer

The vector sum is \((-6, -12)\) with magnitude \(6\sqrt{5}\).

Step 1: Determine Vectors SO and SQ

To find \(\overrightarrow{SO}\) and \(\overrightarrow{SQ}\), we calculate the difference between the coordinates of the points that define each vector. For \(\overrightarrow{SO}\), subtract the coordinates of \(O\) from \(S\), resulting in \(\overrightarrow{SO} = (0-5, 0-9) = (-5, -9)\). For \(\overrightarrow{SQ}\), subtract the coordinates of \(Q\) from \(S\), resulting in \(\overrightarrow{SQ} = (4-5, 6-9) = (-1, -3)\).

Step 2: Add Vectors SO and SQ

Using vector addition, add \(\overrightarrow{SO}\) and \(\overrightarrow{SQ}\) by adding their corresponding components. Thus, \(\overrightarrow{SO} + \overrightarrow{SQ} = (-5 + -1, -9 + -3) = (-6, -12)\).

Step 3: Compute the Magnitude of the Sum

To compute the magnitude of the resulting vector \((-6, -12)\), use the formula for the magnitude of a vector, \(\sqrt{a^2 + b^2}\). Thus, the magnitude is \(\sqrt{(-6)^2 + (-12)^2} = \sqrt{36 + 144} = \sqrt{180}\). Simplifying \(\sqrt{180}\), we get \(6\sqrt{5}\).

Step 4: Use the Parallelogram Law

The parallelogram law states that the sum of two vectors is represented by the diagonal of a parallelogram formed by the vectors. Here, we already computed the vector sum which corresponds to one diagonal in the parallelogram. Visually, if you draw both vectors \(\overrightarrow{SO}\) and \(\overrightarrow{SQ}\) from the same point on graph paper, the sum is denoted by a vector pointing to the opposite corner of the parallelogram.

Key Concepts

Coordinates

Coordinates refer to a set of values that specify a point in space. In a 2D plane, these are expressed as Cartesian coordinates, which consist of two numbers: an x-coordinate and a y-coordinate. Each coordinate pair identifies the precise location of a point from a fixed origin, usually denoted as point O.

For example, the coordinates of point S are (5, 9). This means the point S is located 5 units horizontally and 9 units vertically from the origin O(0,0). When considering vectors, coordinates play a crucial role in determining the direction and magnitude of the vector.

To find a vector like \(\overrightarrow{SO}\), you calculate the difference between the coordinates of point S and the origin O. This is done by subtracting the coordinates of O from those of S to get \(\overrightarrow{SO} = (0-5, 0-9) = (-5, -9)\). This resulting vector indicates the direction and distance from the start point S to the end point O.

Magnitude of a Vector

The magnitude of a vector represents its length and is derived from the Pythagorean theorem. To find the magnitude in a 2D plane, use the formula \(\sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the components of the vector.

For example, consider the vector \((-6, -12)\). Here, \(a = -6\) and \(b = -12\). The magnitude is calculated as follows: \(\sqrt{(-6)^2 + (-12)^2} = \sqrt{36 + 144} = \sqrt{180}\).

Simplifying \(\sqrt{180}\) results in \(6\sqrt{5}\), signifying the vector's length from its starting point to its ending point in space. This measurement is crucial for understanding vector quantities like force or velocity, where the direction and size are both essential.

Parallelogram Law

The parallelogram law is a geometric way to visualize the addition of two vectors. When two vectors originate from the same point, they can form two adjacent sides of a parallelogram. The diagonal that runs from the common point of these vectors represents their vector sum.

For instance, if you have vectors \(\overrightarrow{SO}\) and \(\overrightarrow{SQ}\), start both from the point S. If drawn on graph paper, these vectors create a shape resembling a parallelogram. The vector sum, \(\overrightarrow{SO} + \overrightarrow{SQ}\), corresponds to the diagonal stretching from the starting point, point S, to the opposite corner of this parallelogram.

• This physical representation helps in visualizing how vector directions and magnitudes combine.
• Using the parallelogram law ensures a more intuitive understanding of vector sum and spatial relationships.

The same result, \((-6, -12)\), can be reached through mathematical vector addition, confirming the law's correctness.